Is 3.567 rational?

2 Answers
Rachel
Apr 12, 2018

Yes. Irrational values, like #pi# or #sqrt2#, have a never ending decimal value that does not repeat. But #3.567# does have an end, and we can also write it as a fraction and have it remain accurate: #3567/1000#

There is no decimal or fractional value of #pi# that is accurate (not an approximation but a representation of the exact value of #pi#), so #pi# is irrational

Tony B
Apr 12, 2018

Yes! The decimal equivalent of a #ul("rational number")# is either

Terminating decimal #-> "example " 0.125#
Has a repeating cycle of values for ever#->" example " 0.33333....#

Another example: #" "0.125125125125...#

Explanation:

#color(red)("Just for the hell of it")# lets determine the fractional equivalent of
0.125125125....

Set #x=0.125125125...." "Equation(1)#

We need to 'get rid' of the decimal part so to do this

Multiply both sides by 1000 giving

#1000x=125.125125125...." "Equation(2)#

Subtract Eqn(1) from Eqn(2) to 'get rid' if the decimal

#1000x =125.125125125.....#
#ul(color(white)(1000)x=color(white)(12)0.125125125...larr" Subtract")#
#color(white)(1)999x=125#

Divide both sides by 999

#x=125/999 color(white)("d") =color(white)("d")0.125125125....#

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